This is a survey of various important classical results of M. G. Krein on positive linear operators on an ordered Banach space with modern proofs, including several improvements. The Krein space in this paper means an ordered (real) Banach space with a strong unit. Here a positive element \(u\) is a strong unit if for each element \(x\) there exists an \(\alpha>0\) such that \(\alpha u\geq x\). Here are sample statements. If \(\{T_ \alpha;\;\alpha\in A\}\) is a commuting family of positive operators, then its adjoint family \(\{T_ \alpha';\;\alpha\in A\}\) has a common positive eigenfunctional. If, in addition, all \(T_ \alpha\) make invariant a common strong unit, then some non-zero positive linear functional becomes invariant for all \(T_ \alpha'\). The assumption of the last statement can be modified in the following way; some non-zero element is invariant for all \(T_ \alpha\), and all \(T_ \alpha\) are norm-contractive.